Question:
Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
30
∑ (−3i+5)
i=1
Answer:
The first three terms are : 2, -1 and -4
The last term is: -85
The sum of the sequence is: -1245
Step-by-step explanation:
Given;
==================================
30
∑ (−3i+5) -------------------(i)
i=1
==================================
Where the ith term aₙ is given by;
[tex]a_{i}[/tex] = [tex]-3i + 5[/tex] -------------------(ii)
(a) Therefore, to get the first three terms ([tex]a_1, a_2, a_3[/tex]), we substitute i=1,2 and 3 into equation (ii) as follows;
[tex]a_{1}[/tex] = [tex]-3(1) + 5[/tex] = 2
[tex]a_2 = -3(2) + 5 = -1[/tex]
[tex]a_3 = -3(3) + 5[/tex][tex]= -4[/tex]
Since the sum expression in equation (i) goes from i=1 to 30, then the last term of the sequence is when i = 30. This is given by;
[tex]a_{30} = -3(30) + 5 = -85[/tex]
(b) The sum [tex]s_n[/tex] of an arithmetic sequence is given by;
[tex]s_n = \frac{n}{2}[a_1 + a_n][/tex] -----------------(iii)
Where;
n = number of terms in the sequence = 30
[tex]a_1[/tex] = first term = 2
[tex]a_n[/tex] = last term = -85
Substitute the corresponding values of n, [tex]a_1[/tex] and [tex]a_n[/tex] into equation (iii) as follows;
[tex]s_n = \frac{30}{2}[2 + (-85)][/tex]
[tex]s_n[/tex] = 15[-83]
[tex]s_n[/tex] = -1245
